Electric Waves round a Large Sphere. 167 



and on examination, no further derivate is of higher order 

 than z. Thus to order zero in z 3 from (87) 



IJo+ ^> +U 2 + = „^. 0+ D M€ .*i+DV.|j? + ... 



or (>/r + 2\/r 1 )/2t. This becomes, in terms of v', on 

 reduction, 



and therefore from (84, 85), if (r l5 y 2 )= ±<j>tf, 



-«<«.[("- u r--)(5-S)]; 



so that the term of this order in the magnetic force vanishes. 

 Moreover, it may be shown at once that whenever 



U + — + -^ +... 



is the product of e~i a '' and an odd function of v', the same 

 result must occur. A further examination of the magneti 

 force indicates in this way that the set of terms of order z~ J 

 contributes zero when y"==0, or when the point at which the 

 effect is desired is on the surface of the sphere, so that 

 (j) n =(f) nr . The magnetic force is thus at least two order,? 

 higher than was shown in the last section. 



These results indicate that the vanishing of these terms of 

 successive orders is general, and this will now be shown to 

 be the case, by an independent method. 



Consider u as a function of m or n + j>. It may be shown * 

 that if R, t is derived from Bessel functions of real order m, 

 without any restriction of m to half integral values, 



H 



„ = — f ' K (2z sinh t) cosh 2mt dt. . . (89) 



IT ! . 



This integral is an even function of m, and so also therefore 

 is R n . Moreover, % n is also an even function because 



tan Xn = — iBR«/d~- 

 Thus ?^, which is proportional to mR tt (l + r' x »), is an odd 

 * Phil. Mag. Feb. 1910, p. 234. 



